Officially it varies - the UN doesn't recognise Kosovo, but several others do. The big problem for you, as you observed - is Serbia, which considers it to be another province, and indeed administers five of the municipalities.
The best warning I can find seems to come from Wikitravel:
Serbia officially states that it will block passports containing stamps or visas from Kosovo. However, in practice, immigration officers would usually just cancel the Kosovan stamps and replace them with Serbian ones.
If you are just visiting the region, visit Serbia first. You will not
be given a Serbian exit stamp if you enter Kosovo from Serbia. If you
are living in or intend to travel frequently to Serbia, you should get
matching pairs of entry and exit stamps; this would mean backtracking
and leaving through Serbia via a regular border crossing point. If you
travel a lot in the region, your passport will be crowded with stamps
anyway and border guards may not be able to find the matching stamps,
and normally let you pass. Just avoid mentioning Kosovo. You can ask
Kosovar authorities not to stamp your passport at the border.
So unless you obsess with stamps (which I can empathise with), the simplest solution would be to ask them not to stamp the passport, much like many travellers do with Israel.